Geology Reference
In-Depth Information
Now consider the
prediction error power
of the (
N
+
1)-length prediction error
filter. It is
⎝
1
g
k
f
j
−
k
⎠
⎝
⎠
E
⎩
⎭
N
N
E
j
∗
j
f
∗
j
−
1
g
∗
l
f
∗
j
−
l
P
N
+
1
=
=
f
j
−
k
=
l
=
=
E
⎩
k
=
1
g
k
f
j
−
k
⎭
l
=
1
g
∗
l
f
∗
j
−
l
−
f
∗
j
N
N
f
j
f
∗
j
−
f
j
E
⎩
⎭
N
N
1
g
∗
l
f
∗
j
−
l
+
1
g
k
f
j
−
k
k
=
l
=
l
=
1
g
∗
l
E
f
j
f
∗
j
−
l
N
k
=
1
g
k
E
f
∗
j
f
j
−
k
N
=
φ
ff
(
0
)
−
−
N
N
1
g
k
g
∗
l
E
f
j
−
k
f
∗
j
−
l
.
+
(2.76)
k
=
1
l
=
Thus,
l
=
1
g
∗
l
φ
ff
(
l
)
k
=
1
g
k
φ
∗
ff
(
k
)
N
N
P
N
+
1
=
φ
ff
(0)
−
−
N
N
1
g
k
g
∗
l
φ
ff
(
l
+
−
k
).
(2.77)
k
=
1
l
=
From the unit prediction equations (2.73), the last term on the right-hand side of
expression (2.77) can be transformed to give
l
=
1
g
∗
l
φ
ff
(
l
)
k
=
1
g
k
φ
∗
ff
(
k
)
l
=
1
g
∗
l
φ
ff
(
l
)
N
N
N
=
φ
ff
(
0
)
P
N
+
1
−
−
+
k
=
1
g
k
φ
∗
ff
(
k
)
N
k
=
0
γ
k
φ
∗
ff
(
k
)
N
k
=
0
γ
k
φ
ff
(
N
=
φ
ff
(0)
−
=
=
−
k
).
(2.78)
Writing the unit prediction equations (2.73) in terms of the prediction error coef-
ficients they become
N
0
γ
k
φ
ff
(
m
−
k
)
=
0,
m
=
1,...,
N
.
(2.79)
k
=
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